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In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We prove that the velocity of the relaxing equations converges weakly to the velocity of the relaxed equations, while other variables of the relaxing equations converge strongly to the corresponding variables of the relaxed equations. We prove that as relaxation time approaches 0, there exists an initial layer for the ill-prepared data, the convergence of the velocity is strong outside the layer; while there is no initial layer for the well-prepared data, the convergence of the velocity is strong near t = 0. The strong convergence rates of all variables are also estimated.

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index are interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.

We discuss Nakamaye's Theorem and its recent extension to compact complex manifolds, together with some applications.